Multi-Objective Evolutionary Algorithms with Sliding Window Selection for the Dynamic Chance-Constrained Knapsack Problem

Evolutionary algorithms are particularly effective for optimisation problems with dynamic and stochastic components. We propose multi-objective evolutionary approaches for the knapsack problem with stochastic profits under static and dynamic weight constraints. The chance-constrained problem model allows us to effectively capture the stochastic profits and associate a confidence level to the solutions' profits. We consider a bi-objective formulation that maximises expected profit and minimises variance, which allows optimising the problem independent of a specific confidence level on the profit. We derive a three-objective formulation by relaxing the weight constraint into an additional objective. We consider the GSEMO algorithm with standard and a sliding window-based parent selection to evaluate the objective formulations. Moreover, we modify fitness formulations and algorithms for the dynamic problem variant to store some infeasible solutions to cater to future changes. We conduct experimental investigations on both problems using the proposed problem formulations and algorithms. Our results show that three-objective approaches outperform approaches that use bi-objective formulations, and they further improve when GSEMO uses sliding window selection.